The apparent ``black box'' nature of neural networks is a barrier to adoption in applications where explainability is essential. This paper presents TAME (Trainable Attention Mechanism for Explanations), a method for generating explanation maps with a multi-branch hierarchical attention mechanism. TAME combines a target model's feature maps from multiple layers using an attention mechanism, transforming them into an explanation map. TAME can easily be applied to any convolutional neural network (CNN) by streamlining the optimization of the attention mechanism's training method and the selection of target model's feature maps. After training, explanation maps can be computed in a single forward pass. We apply TAME to two widely used models, i.e. VGG-16 and ResNet-50, trained on ImageNet and show improvements over previous top-performing methods. We also provide a comprehensive ablation study comparing the performance of different variations of TAME's architecture. TAME source code is made publicly available at https://github.com/bmezaris/TAME